Perturbation analysis of a semilinear parabolic problem with nonlinear boundary conditions
type: Item
instance of: scholarly article
title: Perturbation analysis of a semilinear parabolic problem with nonlinear boundary conditions
zbMATH Open document ID: 0858.35066
author: Sabina
author: Domingo Salazar
author: A. A. Lacey
author: John R. Ockendon
published in: Rocky Mountain Journal of Mathematics
publication date: 1997-03-23T00:00:00Z
full work available at URL: contents.htm
review text: The objective of the paper is the study of the nonlinear, parabolic, initial-boundary value problem: \[ u_t=u_{xx}-\lambda u^p, \qquad x\in(0,1), \quad t>0, \] \[ u_x(0,t)=0, \;\;u_x(1,t)=u^q(1,t), \quad t>0, \qquad u(x,0)=u_0(x), \] where \(\lambda\), \(p\), \(q\) are constants, \(\lambda>0\), \(p>1\), \(q>1\) and \(u_0>0\). A formal asymptotic description of the steady states is presented and the stability of these steady states is studied in the case when \(p\) and \(q\) exceed unity. In particular, a perturbation analysis is made taking \(p=1+\varepsilon \alpha\) and \(q=1+\varepsilon\beta\) where \(\varepsilon\) is a small positive parameter. The analysis emphasizes the role of \(\alpha/\beta\) especially when this parameter is near one or a half. The authors expect that their response diagrams will give a reliable guide to the size of stationary solutions for larger values of \(\varepsilon\). They indicate also how the steady states can be used to give information on the solution of the unsteady problem.
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35C20
zbMATH DE Number: 894679
zbMATH Keywords: perturbation analysis
reviewed by: S. Totaro
MaRDI profile type: Publication
OpenAlex ID: W2070582054
cites work: Q3974148
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cites work: Q3480543
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DOI: 10.1216/RMJM/1181072111
description: scientific article; zbMATH DE number 894679